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On Anti-Integrator-Windup and Global Asymptotic Stability
2a-16 2
Copyright © J996IFAC 131h Trienrual World Congress, San Francisco. USA
On Anti-Integrator-Windup and Global Asymptotic Stability Navneet Kapoor\ Andrew R. Teel.... and Prodromos Daoutidis'" University of Minnesota, Minncapolis, MN 55455
.. Departm ent of Ch emical Engin eering and Mate'fit11IJ Science. Emails:kapoorOcc:m . ... mn. eau} dll.outidiOnm!J.umn.edu. Fltnded in par! bll the Shell F01!ndation . •• Department of ElutricaI Enginel!ring. lI.nder gmnt ECS-9309523.
F:·mu.i/: /edOee.1tmn.td1l..
F'lt.ndtd in parI by the NSF
Abstract . This article focuses on critically stable linear sy~d.ems and proposes an approach to address anti-integrator windup that. guarantees ,;lobal asymptol.ic dosed-loop stability and a. d e~~ closed-loop performance . Specifically, given a. dynami.c state or output feedback compensator "containing" integral action, that has b~n designed ignoring ~onBtr&ints, the compensator dynam· ics are modified, .... ia a standard Iln~i·windup modificat.ion, whenever the input is satura.tf!
1. Introduction
All pra.ctical control systems are limited by the capacity of the aduators. It is well recognized (.. e e.g. (Doyl. et al. , 1987» that the ill-etrects of actuator saturat.ion are particularly pronounced whenever the compensat.or possesses slow dynamics, resulting in oscillatory hehav ior, overshoots, limit cycle behavior and even instabihty. These ill-effects are collectively referred to as the problem of '{oin(lup. They hav~ led to F.lUbst.antial re· ~ at(',h on incorporating modificati ons (commonly termed as anti-windup schemes) in a compensator that ha..., been designed without accounting for constraints, such that the dosed-loop behavior is satisfactory even in Lhe presence of constraints. The prevalent approach in the design of antiwindup schemes involves initially designing the co rnpenija tor ignoring constraints and then modi· fying it. only when the input is saturated (see e.g. ( Walgama and Sternby, 1990». Though such an approach ensures recovery of tbe nominal dosedloop performance for "small" signals, it may make it difficult. t.o guarantee acceptable overall behavior} i.e. , to simultaneously guarantee a desired dosed· loop performance and a desired region of closed· loop stabilit.y. This is reflected in the fact that most of the anti-wind up schemes within this approach are ad hoc in nature and do not guarantee asymptotic closed-loop stability (with a desired basin of attraction) even for open-loop HurwiLz systems .
1512
In (Kapoor et al., 1995a) , we developed a novel approach to ant.i-winoup design for general linear systems (i.e., with no restriction!ii on t.he location of the open-loop eigenvalues) that allows for arbitrary fa.c;t attenuation of t.he effect of windup and hence not only guarant.ees a. desirable region of closed-loop stability but also ensures a. desirable closed-loop performance. The key idea behind this approach is to characterize the closedloop behavior under a dynamic feedback controller in terms of the behavior under a static feedback controller. Such a characterization is only na.tural considering the fact that windup is usually associated with dynamic compensators. Some of these results, which arc "regional" in nature, were presented in (Kapoor et al., 1995b) . The cu rrent 'work focuses on critically stable linear systems under compensators t.hat "contain" integral action and presents results tha.t are ';globaI1! in nat.ure. Specifically, given a dynamic sta.te feedback co mpensator of t.he form :
( I) where
Xe
represents the states of the compensator,
ex is the input to the compensator and refers to a. vector of measurements, 'U is the output of the compensa.tor, and Cc and De are constant matrices, the proposed approach involves designing the gains Cc and Dc and modifying Xc (via a standard anti-windup modification) when the manipulated
input hits the cont.rollimits. Not.e that modifying the compensat.or rlynamics only for large signals ensures that the compensator ret.ains its ability to reject constant exogenous signals. The gains D, and L are designed such that the closed-loop system under the dynamic feedha.ck (.ompensator is transformed into a cascade of an asymptoticalJy stable subsystem, under a given static feedback compensator, being forced by the states of an exponentiaIly stable subsystem. This structure allows us to guarantee global asymptotic st.ability for the closed-loop system in the presence of constraints and, moreover, reduce the behavior of the closed-loop system to that under the static feedback controller arbitrarily fast. When all the process states are not available for measurement, the compensat.or dynamics are augmented with an obse rver for the state~ .
era
The rest of the manuscript is structured as follows. In section 2, we present the class of systems we will focus on and a few technical preliminaries. In section 3, wc will present a state feedback result t o the design problem. In section 4, we genernli7.e the statc feedback result to the case of output feedback. In section ,I), we quantify th e error regulation property (i.e., rejecting constant exogenous signals) of the proposed design. Finally, in section 6, we illustrat.f! thf! t.heory t.hrough two examp les.
We will consider linear systems of the form :
y
= Ax+Bu(u)
=
Cx
=..
=
For the system of Eq.2, we will consider a compensator of the form of Eq. l wbose dynamics have been modified by the anti-windup design modification L(u(u) - u) (see e .g. (A.strom and Rundqwist, 1989)). The following assumption is also invoked: Assumption AI: The pair [ 00 CA' j, controllable.
[RO. j.S
In the following section I we will present a procedure to design Cc, Dc and L such that the origin of the system of Eq.2 under the compensa.tor of Eq.l, with the anti-windup modifi catio n, is globally asymptotically stable.
2. Preliminaries
x
For the sake of simplicity, it will be assumed that k l l: is a a-safe feedback for a nonlinearity (T that satisfies assumption AO, then it is also a usafe feedback fo r ,,(u) (or, in other words, (A + Bk t) is Hurwitz). Note that there always exists a u-safe feedback for a stabili:lable, critically st.able system of the form of Eq .2 where u satisfies assumption AO. The following lemma establiBhe~ the existence of one such feedback: Lemma 1: If the pair (A, B) is stabi/izabic alld AT p + PAS: 0 for some pO.'jitiTlf. de.finite symmetric matrix P then u _ET Px is a u-safe feedback for the system i: = Ax + Bu(u), where u satisfie.s assumption AO. In the special case of A being Hurwit1. , any feedback that globally asymptotically stabilizes the origin of the system of Eq.2 is a tT-safe feedba ck (under assumption AO) .
if
(2)
where x E !Rn represent the state variables , U E IR..m represents t he vector of manipulated inputs, y E rn.m refers to a vector of measurements , A , B and C are constant matrices of appropriate dimensions and it is assumed that A is criti cally stable (but not necessarily Hurwit.z), i.e., there exist.s a positive definite symmetric matrix P such that AT P + P A <: O. We also make the following standing assumpt.ion on the function u , whi ch refers to a. general input nonlinearity: Assumption AO: The function u : IRm _ JR.m is globally Lipschitz, bounded and satisfies th e prop~ erty that uT u(u) > 0, 1t # O. Note that assumption AO is satisfied by the stan~ dard J;at uratio n nonlin carit.y and the tanh fun cLio n am.o ng ot hers. 'Ve now postulate th e following notion : D efinition 2.1 ff-fiafe feedback: Th e Jee.db(J.(;k u = k1z is u-safe for the syst em of Eq.2 if it guarant ees that, for each :1: 0 E lRn , thr; trajectory a/the syst em;i; = Ax+Bu(k 1 x+d), x(O) = Xo converges to ZeT'U Jor all 8Tnoolh slgnals cl lhat converge exponentially to zero.
1513
3. State Feedback Design The system of Eq.2 under the compensator ofEq.l subject to the anti-windup modification has the following form:
Cx + L[IT( Cox,
+ D,,,,)
-(C,x, + D,x)] ;i;
Ax + Bu(C,x ,
(3)
+ D, x).
We are now in a positio n to prese nt. the main result of this section in t.he form of the following theorem : Theorem 1: Consider th e .~yst e m of Eq.3 f01' which. assumption Ai holds true . 1f k1x is a usafe for the system of Eq.2, then choosing Cc such that -C(A + Bk, )-' BC, is Hurw;t" D, = k, - C,C(A + Rk,)-' and L = C(A + IJk,) - ' B guarantees that the origin of th e sy.~tem of F:q.3 i" GAS. Proof: Consider the change of variables z :;;; x, - C(A + Bk,J-1X. The closed-loop system in the z, x coordinates has the following form for the
above choices of Dc and L:
= ;; =
Bk,)-' (x - x ), the system of Eq .6 assumes t.he following cascaded structure:
-C(A + Bkt)-' BC,z
i
Ax
+ B
(4 )
Since - C IA + Bk, )-' BC, is Hurwitz (a choice of Cc that guarantees this always exists by assum pt ion Al), z decays exponentially to zero. Moreover , siuce k1 x is a u-safe feedback for th e syst.em of Eq.2, it follows that x decays to zero. Hence, the system of Eq.3 is GAS. 4. Output Feedback Design When a ll the process states are lIO t. availa ble for measurement, we augment the dynami cs of the compensa tor of Eq.l with a state observer . To this end , let us consider the following dyna mic o utput feedback compensator : i:
=
Ai + B(C, x,
+ Dcx)+
H(Ci: - Cx )
u
(5)
ex
.i ~.
=
Ccxc
+ Dei
where i refers to the observer states and if is chosen such that (A + He) is Hurwit.z, un der the ass umption that: Assumption A2: The pair (C, A ) is ob.servable. We will modify the compensator dynamics with the t erm L[u(lt ) - uJ , where L
= [i~ 1, such
tha t the closed-loop system assumes t he following form:
i:
=
Ai: + B (Cer,
+ D,i)
+H(Ci:- Cx) +L,[u(C,x,
-(Cox,
+ D,i )
+ D,x)J
Cx + L,[u(C,x, -(C, x,
(6)
+ HC )i -CIA + Bk, )-' B C,z+ C(A + Bk,J- ' Bk, x = Ax + B
i
z x
Since z and x decay exponentially t o zero and k 1 x is a u-safe feedback for t.he syste m of l!;q.2, it follows that x decays to zero. Hence , the Elystem of Eq.7 is GAS. Remark 1: From the cascaded stru ctures of Eq .4 and Eq. 7) onc can conclude that as the z a nd x subsystems decay to t he origin, the behavior of the closed-loop systems reduces to that under the feedback ktX' . Hence, by placing t he eigenv
5. Error Regulation vi a the Proposed Anti-Windup Design In this section, we will quant ify the a.bility of the state compensator design presented in section 3 to reject "small" e xogenoll ~ signal8 (i .e., to achieve restricted set-poiut t racking and disturbance regulation) . A similar analysis can be cartied out for t he ou t put feedback case but will be omitted here for the sake of brevity. We cast the system ofEq .2 in the framework of the linear regulator theory and consider th e following system:
+ D, i)
Ax + B
+ D,i)J
and design Cc , D c,L} , L z such that the origin of the system of Eq .6 is GAS . The main result of this sec tion is stated in the following theorem : Theorem 2: CQn~ide r u syste. m of th e form of Eq.6, s. ch that assumptions A 1 and A 2 hold lrue. If ( A + HC) is EluTWit . and k, x is a
w
o
e
C x +Qw
=-
and f J2 B glUlrantees that th e origin of th e system of Eg. 6 is GAS. Proof: Defining ;; = x -- x, • = x , - e (A +
1514
+ Rw (8)
where w E IRP represents constant unmeasured signals affecting the system , e E m m represents the error that needs to be regulated . Rand Q are constant mat rices of appropria te dimensions. Let us now consider a. controller of t he form :
e + L[u(C, x, + D, x) -(Cox, + D, x )J
s af e f eedback f or th e syste'm of Eq.2, th en cho osing C, su ch Ihat -C(A + Bk , )-' BC, is HU TWit z,
D, = k ,-C,C(A+Bk,)-', L, =C( A+ Bk, )-' B
(7)
u
(9 )
Ccxe + Dcx.
Note that the input. to the. compensat or is the error e instead of the output ,rector C x . It will be
assumed that assumption Al holds true for the system ofEq.8 under t.he compensator of Eq.9 and th a t C c. D c and L have been designed in accord a nce with the design procedure outlined in se<;tion 3. In that case there always exists 8. matrix (sec e.g. (Desoer and Wang , 1980» II = [
~:
1
which solves the linear matrix equations:
All,
+ B(C,ll, + D,ll,) + R = Cll,
+Q
=
0 O.
(10)
Tn order to simplify the analysis, we defin e 'lde_
viatlon" vanables ITI W,
T}2
=x -
TT
= [ ~~
l'
with 7]1
= Xc -
II 2 w. Since the gains C c, Dc and
L have been designed in ac.cordance with the pro. cedure presented in section 3, by employing the change of variables z = 'I' - C(A + BkJ)-' 1/2, the system of Eq_8 under the compensator of Eq.9 can be transformed into the following form : z
=
-C(A
~,
=
Ary,+
+ Bk,)-' BC,z (11)
B[<1(k'1]'
+ C,z + fw)
=
- fw].
= = ,-= = O. To this end,
(C,ll, + D,ll,). Note that e C1J,. where I' Hen ce , if we can ensure that HIll 1]2 0, t.h en we a re guaranteed that lim f k l 7J2 is a. O'-safe feedback for the system:
,-=
~,
AT!, + B [a(k,,!,
+ fw) -
rwJ
if
(12)
then we are guaranteed that TJ2 decays asymptotica.l1y La the origin . It can easily be verified that k, _8T P (cho""n as the <1-safe feedback given in lemma 1) is a O"-safe feedback for the above syst em provided that :
=
6. Simu1ation Studies We now apply the theory to two examples. The first example considers an opell ~ loop Hurwitz 81S0 system subject to input constraints. The control of this system under PIn controllers has been extensively st udied in (Astrom and Rllndqwist , 1989) and windup has heen reported as a major problem. To this end , classical antiwind up design modifications have hee n incorporated in the dosed-loop and a significant improvement in performance over the case of no windup compensation has heen seen. However, it has not been established that these schemes guarantee asymptotic closed-loop stability. For this example, we will employ the state and outpu t feedback compensator designs presented in section 3 and illustrate that the proposed anti-wind up design not only guarantees global asymptotic closed-loop stability but also enforces a desirable closed-loop performa.nce. The second exa.mple considers an openloop HUTwitz MIMO system t.ha t if:! subject to a pulse disturbance. it has been obse rved (Campo and Morari, 1990) that asympt otic stability may not be guaranteed for this system und er a PI controller with a classical ant.i-windup scheme, and onc may have to incorporate directionaJity compensation in order to do so. For this example, it will be shown that the proposed anti· wind up design guarantees global asymptoti c stability a nd a desirable closed-loop performance without having to incorporate directionality comp ensation. EX81nple 1: This example considers a ca.scade of two identical tanks, whose dynami cs are modeled by tbe following set of differential equations (Astrom and Rundqwist, 1989):
X, X, tU
for k,1J2 cF O. In the special case of (1 being the standard decentralized saturation fun ction sat wiLh
(14)
=
1~ 1, .. ' . rn , the bounds on fw under which the condition of Eq.13 holds true are:
(15) For A being Hurwitz, it is straightforward to see that t.he inequalit.y of Eq.15 can be relaxed to:
(16)
1515
e
-O.015x, + O.05.at(u ) + p ,lII 0.015x, - 0.015x , O,w(O) 1
=
(17)
X2 - P2'W
The system is open-loop Hurwitz with repeated eigenvalues at -0.015. The manipulated input u E IR is the influent flow rate to the upper tank and sat refers to the standard saturation nonlinearity described by Eq.14, with tt m a.t' = 1 and Umin = O. The states Xl and X 2 a re the upper and lower tank levels respectively, w refers to exogenous signals affec.ting the process and e is the error that needs to be r~gula.f,ed . At Lime t = 0, with XI = X 2 = 0, a se t~ point change, P2W = t, is introduced. At t 250 , the level of the lower lank is subject to an impulse di sturbance of magnitude 0.5. At t ime t = 500, a load dist urbance Pl W = -0.0325 is introdu ced on the upper tank. For the system of Eq .17 , we employed the proposed state and output feedback compensator designs so as to steer e to th e origin. The
=
gain kl was chosen as kl = [-0 ,9167 - 0.4167) ('"' as t.o globally asymptotically stabilize the system of Eq,17 with w = 0) and Co was designed as -20 . Following the analysis in section 3.3, it can be verified that rw = 0 .3 for t < 500 and rw 0 .95 for t > 500. Thus, the e-;cor e will decay to the origin , while the process states stay b ounded . For the output feedback design , we assumed that only X:J is available for measurement. The matrix H was designed such that the eigenvalues of (A + He) are at -2,5 each_ Fig,1 shows the errur a.nd the manipulated input profiles under t.h e stat.e feedback compensator, whereas Fig.2 shows the corresponding profiles under t he o utput feedback compensator. In Figs .l and 2, we a lso show for the sake of comparison the error and the manipulated input profiles under a PID controller (see (Astrom and Rundqwist , 1989») wi t ho ut any an t i-windup compensation. Note t hat the oscillations a.nd overshoot.s due to windup have been completely attenuated under th e proposed design . For a. comparison with the profiles under t he convent ional anti-windup schemes please refer to (Ast rom and Rundqwist , 1989) or (Kapoor et ai"~ 1995.),
=
Example 2: This example considers a twodim ensional '
z,
= W = el = e, =
;"
-O,lx, + 0,5.ot,(u,)+ 0.48at, (u, ) + PIW - O_l x , + OAsat l ( UI)+ 0 ,3sat , ( ",) + p,w O, w(O) 1 XI
(18)
=
=
' d as: Ce SIze
30 1( h = [2,5 -30 -17.5 note t at
Xc
E IR')
and De and L were accordingly computed. It can clearly be checked that for 0 :S t :S 5, r,w = 2, r 2 tJ! = -4, and for t > 5, rIw 0; 2w clearly then, the error e will decay to the origin. FigA shows the state and the m anipulat ed input profiles under the dynamic comp ensator. As can be seen, the states stahilize at the origin wit h negligible effects of windup.
=r
=
T he two examples clearly reiterate the desirable features of guaranteed stability and satisfactory performance ensured by the proposed design approach.
7. Conclusi ons In this note we have presented an ant i-windup approach for critically stable systems that allows us to completely attenuate the effect of windup, while guaranteeing global asymptotic stability. In (Kapoor et ai"~ 1995a), we have ge neralized these rffi ults to general linear systems under more general linear compensators. Future research will f~ cus on developing "optimal" designs for th e a ntiwindup gain matrix with the restri ction of recovering the nominal closed-loop behavior whenever t he inpu t is not saturated .
x~
The system is open-loop HUfwitz wi t h repea ted eigenvalues at -0.1 and satl and sat2 refer t o th e stan dard saturation nonlinearity wi t h ui QZ :;:;:: 3, ul in = -3 and u2'ax :;:;:: 10, u 2 ,n -10 . At tim e t 0 pulse changes, PI wand P2W of dura.tion 5sec and magnitudes 0.6 and 0.4 a re introd uced in the Xl and the X2 subsystems respectively. This example has been consi dered in (Cam po and Morari, 1990) where these pulse changes were treated as set-p oint chan ges; here we treat them as unmeasured disturbances. Fig .3 s hows t he st a te a nd t.ht.~ input. profiles un de r a PI controller (designed in (Campo and Morari , 1990)) in th e presen('.e of constraints. A~ can he seen, both t he inputs saturate and drive t he states away from their nominal values. For this example it has been observed (Campo a nd Morari , 1990) t hat a conventional ant i-windup scheme may not be able to guarantee asymptoti c stability without direct ionality compensation.
=
satm ba.."ied on t he procedure outlined in se<: tion 3. The feedba('.k gain kl was chosen as k, -2,5 - 2 ] (so AA to globally asymptotically [ -2 _ 1.5 stabilize the origin of th e system of Eq.1 8 with w = 0) . Based on t his choice, Cc. was synthe-
=
For the system of Eq.1S, we designed a compen-
1516
8, REFEREN CES
Astrom , KJ, and L Rundqwist (l989) , Integrator windup and how to avoid it . Proe. of Amer. Contc_ Con! pp, 1693- 1698 , Campo, P,J, and M, Morari (1990) , Robust control of processes subject to saturation nonlinearities, Camp.t, chem, £ ngng , 14, 343- 358, Desoer, C,A_ and Y,T, Wang (1980), Linear timeinvariant robust servomec ha nism prob lem : A self-contained exposition . C ontrol and Dynamic System, 16 , 81-129, Doyle, J .C" R ,S, Smith and D ,F, Enns (1987) . Control of plants with input ~:\t ura.ti o n nonlinearities. Proc. oJ A m er. C 011tr. Conf pp . 10341039, Esfandiari , F, and Il.K , Khalil (1992), Output feedback stabilization of full y lincariz able systems, Int, J, of Contc, 56 , 1007- 1037_ Kapoor, N" A,R, Teel and p , Daoutidis (1995.), Guidelines for anti-wind up design for linear systems . .':mbmilled lo Auiomatica.
i'
: e
u:l\ :1 './
i
'.'
Time (sec.)
Time (sec.)
Fig.I. State Feedback Design (solid). PI Controller (dash)
I!
::D:' ' .. Q~
;
0.
,
:,
,
U
:
::,i -/V~~
Time (sec.)
Time (sec.)
Fig.2. Output Feedback Design (solid), PI Controller (dash)
Time
Time
Fig.3. PI Controller 0 - , --,-·_-
_---_-~
Uj
X2
'. Time
Fig.4. State Feedback Design
Kapoor, N " p, Daoutidis and A.R, Teel (1995b), Anti-windup with guaranteed stability for linear systems with input. con8iraints. Annual AIChE Mee/mg, paper' no, 178g. Teel, A.R, and L. Praly (1995), Tools for semi, global stabilization by partial state and output feedback, SIAM j, of Conir, and Optimlz. 33,1443-1488. Walgama, K.S. and J, Sternby (1990), Inherent observer property in a class of anti-windup compensators. Int. J. ContT'. 52, 705-724.
1517
Time
_ _ ____,
Copyright © J996IFAC 131h Trienrual World Congress, San Francisco. USA
On Anti-Integrator-Windup and Global Asymptotic Stability Navneet Kapoor\ Andrew R. Teel.... and Prodromos Daoutidis'" University of Minnesota, Minncapolis, MN 55455
.. Departm ent of Ch emical Engin eering and Mate'fit11IJ Science. Emails:kapoorOcc:m . ... mn. eau} dll.outidiOnm!J.umn.edu. Fltnded in par! bll the Shell F01!ndation . •• Department of ElutricaI Enginel!ring. lI.nder gmnt ECS-9309523.
F:·mu.i/: /edOee.1tmn.td1l..
F'lt.ndtd in parI by the NSF
Abstract . This article focuses on critically stable linear sy~d.ems and proposes an approach to address anti-integrator windup that. guarantees ,;lobal asymptol.ic dosed-loop stability and a. d e~~ closed-loop performance . Specifically, given a. dynami.c state or output feedback compensator "containing" integral action, that has b~n designed ignoring ~onBtr&ints, the compensator dynam· ics are modified, .... ia a standard Iln~i·windup modificat.ion, whenever the input is satura.tf!
1. Introduction
All pra.ctical control systems are limited by the capacity of the aduators. It is well recognized (.. e e.g. (Doyl. et al. , 1987» that the ill-etrects of actuator saturat.ion are particularly pronounced whenever the compensat.or possesses slow dynamics, resulting in oscillatory hehav ior, overshoots, limit cycle behavior and even instabihty. These ill-effects are collectively referred to as the problem of '{oin(lup. They hav~ led to F.lUbst.antial re· ~ at(',h on incorporating modificati ons (commonly termed as anti-windup schemes) in a compensator that ha..., been designed without accounting for constraints, such that the dosed-loop behavior is satisfactory even in Lhe presence of constraints. The prevalent approach in the design of antiwindup schemes involves initially designing the co rnpenija tor ignoring constraints and then modi· fying it. only when the input is saturated (see e.g. ( Walgama and Sternby, 1990». Though such an approach ensures recovery of tbe nominal dosedloop performance for "small" signals, it may make it difficult. t.o guarantee acceptable overall behavior} i.e. , to simultaneously guarantee a desired dosed· loop performance and a desired region of closed· loop stabilit.y. This is reflected in the fact that most of the anti-wind up schemes within this approach are ad hoc in nature and do not guarantee asymptotic closed-loop stability (with a desired basin of attraction) even for open-loop HurwiLz systems .
1512
In (Kapoor et al., 1995a) , we developed a novel approach to ant.i-winoup design for general linear systems (i.e., with no restriction!ii on t.he location of the open-loop eigenvalues) that allows for arbitrary fa.c;t attenuation of t.he effect of windup and hence not only guarant.ees a. desirable region of closed-loop stability but also ensures a. desirable closed-loop performance. The key idea behind this approach is to characterize the closedloop behavior under a dynamic feedback controller in terms of the behavior under a static feedback controller. Such a characterization is only na.tural considering the fact that windup is usually associated with dynamic compensators. Some of these results, which arc "regional" in nature, were presented in (Kapoor et al., 1995b) . The cu rrent 'work focuses on critically stable linear systems under compensators t.hat "contain" integral action and presents results tha.t are ';globaI1! in nat.ure. Specifically, given a dynamic sta.te feedback co mpensator of t.he form :
( I) where
Xe
represents the states of the compensator,
ex is the input to the compensator and refers to a. vector of measurements, 'U is the output of the compensa.tor, and Cc and De are constant matrices, the proposed approach involves designing the gains Cc and Dc and modifying Xc (via a standard anti-windup modification) when the manipulated
input hits the cont.rollimits. Not.e that modifying the compensat.or rlynamics only for large signals ensures that the compensator ret.ains its ability to reject constant exogenous signals. The gains D, and L are designed such that the closed-loop system under the dynamic feedha.ck (.ompensator is transformed into a cascade of an asymptoticalJy stable subsystem, under a given static feedback compensator, being forced by the states of an exponentiaIly stable subsystem. This structure allows us to guarantee global asymptotic st.ability for the closed-loop system in the presence of constraints and, moreover, reduce the behavior of the closed-loop system to that under the static feedback controller arbitrarily fast. When all the process states are not available for measurement, the compensat.or dynamics are augmented with an obse rver for the state~ .
era
The rest of the manuscript is structured as follows. In section 2, we present the class of systems we will focus on and a few technical preliminaries. In section 3, wc will present a state feedback result t o the design problem. In section 4, we genernli7.e the statc feedback result to the case of output feedback. In section ,I), we quantify th e error regulation property (i.e., rejecting constant exogenous signals) of the proposed design. Finally, in section 6, we illustrat.f! thf! t.heory t.hrough two examp les.
We will consider linear systems of the form :
y
= Ax+Bu(u)
=
Cx
=..
=
For the system of Eq.2, we will consider a compensator of the form of Eq. l wbose dynamics have been modified by the anti-windup design modification L(u(u) - u) (see e .g. (A.strom and Rundqwist, 1989)). The following assumption is also invoked: Assumption AI: The pair [ 00 CA' j, controllable.
[RO. j.S
In the following section I we will present a procedure to design Cc, Dc and L such that the origin of the system of Eq.2 under the compensa.tor of Eq.l, with the anti-windup modifi catio n, is globally asymptotically stable.
2. Preliminaries
x
For the sake of simplicity, it will be assumed that k l l: is a a-safe feedback for a nonlinearity (T that satisfies assumption AO, then it is also a usafe feedback fo r ,,(u) (or, in other words, (A + Bk t) is Hurwitz). Note that there always exists a u-safe feedback for a stabili:lable, critically st.able system of the form of Eq .2 where u satisfies assumption AO. The following lemma establiBhe~ the existence of one such feedback: Lemma 1: If the pair (A, B) is stabi/izabic alld AT p + PAS: 0 for some pO.'jitiTlf. de.finite symmetric matrix P then u _ET Px is a u-safe feedback for the system i: = Ax + Bu(u), where u satisfie.s assumption AO. In the special case of A being Hurwit1. , any feedback that globally asymptotically stabilizes the origin of the system of Eq.2 is a tT-safe feedba ck (under assumption AO) .
if
(2)
where x E !Rn represent the state variables , U E IR..m represents t he vector of manipulated inputs, y E rn.m refers to a vector of measurements , A , B and C are constant matrices of appropriate dimensions and it is assumed that A is criti cally stable (but not necessarily Hurwit.z), i.e., there exist.s a positive definite symmetric matrix P such that AT P + P A <: O. We also make the following standing assumpt.ion on the function u , whi ch refers to a. general input nonlinearity: Assumption AO: The function u : IRm _ JR.m is globally Lipschitz, bounded and satisfies th e prop~ erty that uT u(u) > 0, 1t # O. Note that assumption AO is satisfied by the stan~ dard J;at uratio n nonlin carit.y and the tanh fun cLio n am.o ng ot hers. 'Ve now postulate th e following notion : D efinition 2.1 ff-fiafe feedback: Th e Jee.db(J.(;k u = k1z is u-safe for the syst em of Eq.2 if it guarant ees that, for each :1: 0 E lRn , thr; trajectory a/the syst em;i; = Ax+Bu(k 1 x+d), x(O) = Xo converges to ZeT'U Jor all 8Tnoolh slgnals cl lhat converge exponentially to zero.
1513
3. State Feedback Design The system of Eq.2 under the compensator ofEq.l subject to the anti-windup modification has the following form:
Cx + L[IT( Cox,
+ D,,,,)
-(C,x, + D,x)] ;i;
Ax + Bu(C,x ,
(3)
+ D, x).
We are now in a positio n to prese nt. the main result of this section in t.he form of the following theorem : Theorem 1: Consider th e .~yst e m of Eq.3 f01' which. assumption Ai holds true . 1f k1x is a usafe for the system of Eq.2, then choosing Cc such that -C(A + Bk, )-' BC, is Hurw;t" D, = k, - C,C(A + Rk,)-' and L = C(A + IJk,) - ' B guarantees that the origin of th e sy.~tem of F:q.3 i" GAS. Proof: Consider the change of variables z :;;; x, - C(A + Bk,J-1X. The closed-loop system in the z, x coordinates has the following form for the
above choices of Dc and L:
= ;; =
Bk,)-' (x - x ), the system of Eq .6 assumes t.he following cascaded structure:
-C(A + Bkt)-' BC,z
i
Ax
+ B
(4 )
Since - C IA + Bk, )-' BC, is Hurwitz (a choice of Cc that guarantees this always exists by assum pt ion Al), z decays exponentially to zero. Moreover , siuce k1 x is a u-safe feedback for th e syst.em of Eq.2, it follows that x decays to zero. Hence, the system of Eq.3 is GAS. 4. Output Feedback Design When a ll the process states are lIO t. availa ble for measurement, we augment the dynami cs of the compensa tor of Eq.l with a state observer . To this end , let us consider the following dyna mic o utput feedback compensator : i:
=
Ai + B(C, x,
+ Dcx)+
H(Ci: - Cx )
u
(5)
ex
.i ~.
=
Ccxc
+ Dei
where i refers to the observer states and if is chosen such that (A + He) is Hurwit.z, un der the ass umption that: Assumption A2: The pair (C, A ) is ob.servable. We will modify the compensator dynamics with the t erm L[u(lt ) - uJ , where L
= [i~ 1, such
tha t the closed-loop system assumes t he following form:
i:
=
Ai: + B (Cer,
+ D,i)
+H(Ci:- Cx) +L,[u(C,x,
-(Cox,
+ D,i )
+ D,x)J
Cx + L,[u(C,x, -(C, x,
(6)
+ HC )i -CIA + Bk, )-' B C,z+ C(A + Bk,J- ' Bk, x = Ax + B
i
z x
Since z and x decay exponentially t o zero and k 1 x is a u-safe feedback for t.he syste m of l!;q.2, it follows that x decays to zero. Hence , the Elystem of Eq.7 is GAS. Remark 1: From the cascaded stru ctures of Eq .4 and Eq. 7) onc can conclude that as the z a nd x subsystems decay to t he origin, the behavior of the closed-loop systems reduces to that under the feedback ktX' . Hence, by placing t he eigenv
5. Error Regulation vi a the Proposed Anti-Windup Design In this section, we will quant ify the a.bility of the state compensator design presented in section 3 to reject "small" e xogenoll ~ signal8 (i .e., to achieve restricted set-poiut t racking and disturbance regulation) . A similar analysis can be cartied out for t he ou t put feedback case but will be omitted here for the sake of brevity. We cast the system ofEq .2 in the framework of the linear regulator theory and consider th e following system:
+ D, i)
Ax + B
+ D,i)J
and design Cc , D c,L} , L z such that the origin of the system of Eq .6 is GAS . The main result of this sec tion is stated in the following theorem : Theorem 2: CQn~ide r u syste. m of th e form of Eq.6, s. ch that assumptions A 1 and A 2 hold lrue. If ( A + HC) is EluTWit . and k, x is a
w
o
e
C x +Qw
=-
and f J2 B glUlrantees that th e origin of th e system of Eg. 6 is GAS. Proof: Defining ;; = x -- x, • = x , - e (A +
1514
+ Rw (8)
where w E IRP represents constant unmeasured signals affecting the system , e E m m represents the error that needs to be regulated . Rand Q are constant mat rices of appropria te dimensions. Let us now consider a. controller of t he form :
e + L[u(C, x, + D, x) -(Cox, + D, x )J
s af e f eedback f or th e syste'm of Eq.2, th en cho osing C, su ch Ihat -C(A + Bk , )-' BC, is HU TWit z,
D, = k ,-C,C(A+Bk,)-', L, =C( A+ Bk, )-' B
(7)
u
(9 )
Ccxe + Dcx.
Note that the input. to the. compensat or is the error e instead of the output ,rector C x . It will be
assumed that assumption Al holds true for the system ofEq.8 under t.he compensator of Eq.9 and th a t C c. D c and L have been designed in accord a nce with the design procedure outlined in se<;tion 3. In that case there always exists 8. matrix (sec e.g. (Desoer and Wang , 1980» II = [
~:
1
which solves the linear matrix equations:
All,
+ B(C,ll, + D,ll,) + R = Cll,
+Q
=
0 O.
(10)
Tn order to simplify the analysis, we defin e 'lde_
viatlon" vanables ITI W,
T}2
=x -
TT
= [ ~~
l'
with 7]1
= Xc -
II 2 w. Since the gains C c, Dc and
L have been designed in ac.cordance with the pro. cedure presented in section 3, by employing the change of variables z = 'I' - C(A + BkJ)-' 1/2, the system of Eq_8 under the compensator of Eq.9 can be transformed into the following form : z
=
-C(A
~,
=
Ary,+
+ Bk,)-' BC,z (11)
B[<1(k'1]'
+ C,z + fw)
=
- fw].
= = ,-= = O. To this end,
(C,ll, + D,ll,). Note that e C1J,. where I' Hen ce , if we can ensure that HIll 1]2 0, t.h en we a re guaranteed that lim f k l 7J2 is a. O'-safe feedback for the system:
,-=
~,
AT!, + B [a(k,,!,
+ fw) -
rwJ
if
(12)
then we are guaranteed that TJ2 decays asymptotica.l1y La the origin . It can easily be verified that k, _8T P (cho""n as the <1-safe feedback given in lemma 1) is a O"-safe feedback for the above syst em provided that :
=
6. Simu1ation Studies We now apply the theory to two examples. The first example considers an opell ~ loop Hurwitz 81S0 system subject to input constraints. The control of this system under PIn controllers has been extensively st udied in (Astrom and Rllndqwist , 1989) and windup has heen reported as a major problem. To this end , classical antiwind up design modifications have hee n incorporated in the dosed-loop and a significant improvement in performance over the case of no windup compensation has heen seen. However, it has not been established that these schemes guarantee asymptotic closed-loop stability. For this example, we will employ the state and outpu t feedback compensator designs presented in section 3 and illustrate that the proposed anti-wind up design not only guarantees global asymptotic closed-loop stability but also enforces a desirable closed-loop performa.nce. The second exa.mple considers an openloop HUTwitz MIMO system t.ha t if:! subject to a pulse disturbance. it has been obse rved (Campo and Morari, 1990) that asympt otic stability may not be guaranteed for this system und er a PI controller with a classical ant.i-windup scheme, and onc may have to incorporate directionaJity compensation in order to do so. For this example, it will be shown that the proposed anti· wind up design guarantees global asymptoti c stability a nd a desirable closed-loop performance without having to incorporate directionality comp ensation. EX81nple 1: This example considers a ca.scade of two identical tanks, whose dynami cs are modeled by tbe following set of differential equations (Astrom and Rundqwist, 1989):
X, X, tU
for k,1J2 cF O. In the special case of (1 being the standard decentralized saturation fun ction sat wiLh
(14)
=
1~ 1, .. ' . rn , the bounds on fw under which the condition of Eq.13 holds true are:
(15) For A being Hurwitz, it is straightforward to see that t.he inequalit.y of Eq.15 can be relaxed to:
(16)
1515
e
-O.015x, + O.05.at(u ) + p ,lII 0.015x, - 0.015x , O,w(O) 1
=
(17)
X2 - P2'W
The system is open-loop Hurwitz with repeated eigenvalues at -0.015. The manipulated input u E IR is the influent flow rate to the upper tank and sat refers to the standard saturation nonlinearity described by Eq.14, with tt m a.t' = 1 and Umin = O. The states Xl and X 2 a re the upper and lower tank levels respectively, w refers to exogenous signals affec.ting the process and e is the error that needs to be r~gula.f,ed . At Lime t = 0, with XI = X 2 = 0, a se t~ point change, P2W = t, is introduced. At t 250 , the level of the lower lank is subject to an impulse di sturbance of magnitude 0.5. At t ime t = 500, a load dist urbance Pl W = -0.0325 is introdu ced on the upper tank. For the system of Eq .17 , we employed the proposed state and output feedback compensator designs so as to steer e to th e origin. The
=
gain kl was chosen as kl = [-0 ,9167 - 0.4167) ('"' as t.o globally asymptotically stabilize the system of Eq,17 with w = 0) and Co was designed as -20 . Following the analysis in section 3.3, it can be verified that rw = 0 .3 for t < 500 and rw 0 .95 for t > 500. Thus, the e-;cor e will decay to the origin , while the process states stay b ounded . For the output feedback design , we assumed that only X:J is available for measurement. The matrix H was designed such that the eigenvalues of (A + He) are at -2,5 each_ Fig,1 shows the errur a.nd the manipulated input profiles under t.h e stat.e feedback compensator, whereas Fig.2 shows the corresponding profiles under t he o utput feedback compensator. In Figs .l and 2, we a lso show for the sake of comparison the error and the manipulated input profiles under a PID controller (see (Astrom and Rundqwist , 1989») wi t ho ut any an t i-windup compensation. Note t hat the oscillations a.nd overshoot.s due to windup have been completely attenuated under th e proposed design . For a. comparison with the profiles under t he convent ional anti-windup schemes please refer to (Ast rom and Rundqwist , 1989) or (Kapoor et ai"~ 1995.),
=
Example 2: This example considers a twodim ensional '
z,
= W = el = e, =
;"
-O,lx, + 0,5.ot,(u,)+ 0.48at, (u, ) + PIW - O_l x , + OAsat l ( UI)+ 0 ,3sat , ( ",) + p,w O, w(O) 1 XI
(18)
=
=
' d as: Ce SIze
30 1( h = [2,5 -30 -17.5 note t at
Xc
E IR')
and De and L were accordingly computed. It can clearly be checked that for 0 :S t :S 5, r,w = 2, r 2 tJ! = -4, and for t > 5, rIw 0; 2w clearly then, the error e will decay to the origin. FigA shows the state and the m anipulat ed input profiles under the dynamic comp ensator. As can be seen, the states stahilize at the origin wit h negligible effects of windup.
=r
=
T he two examples clearly reiterate the desirable features of guaranteed stability and satisfactory performance ensured by the proposed design approach.
7. Conclusi ons In this note we have presented an ant i-windup approach for critically stable systems that allows us to completely attenuate the effect of windup, while guaranteeing global asymptotic stability. In (Kapoor et ai"~ 1995a), we have ge neralized these rffi ults to general linear systems under more general linear compensators. Future research will f~ cus on developing "optimal" designs for th e a ntiwindup gain matrix with the restri ction of recovering the nominal closed-loop behavior whenever t he inpu t is not saturated .
x~
The system is open-loop HUfwitz wi t h repea ted eigenvalues at -0.1 and satl and sat2 refer t o th e stan dard saturation nonlinearity wi t h ui QZ :;:;:: 3, ul in = -3 and u2'ax :;:;:: 10, u 2 ,n -10 . At tim e t 0 pulse changes, PI wand P2W of dura.tion 5sec and magnitudes 0.6 and 0.4 a re introd uced in the Xl and the X2 subsystems respectively. This example has been consi dered in (Cam po and Morari, 1990) where these pulse changes were treated as set-p oint chan ges; here we treat them as unmeasured disturbances. Fig .3 s hows t he st a te a nd t.ht.~ input. profiles un de r a PI controller (designed in (Campo and Morari , 1990)) in th e presen('.e of constraints. A~ can he seen, both t he inputs saturate and drive t he states away from their nominal values. For this example it has been observed (Campo a nd Morari , 1990) t hat a conventional ant i-windup scheme may not be able to guarantee asymptoti c stability without direct ionality compensation.
=
satm ba.."ied on t he procedure outlined in se<: tion 3. The feedba('.k gain kl was chosen as k, -2,5 - 2 ] (so AA to globally asymptotically [ -2 _ 1.5 stabilize the origin of th e system of Eq.1 8 with w = 0) . Based on t his choice, Cc. was synthe-
=
For the system of Eq.1S, we designed a compen-
1516
8, REFEREN CES
Astrom , KJ, and L Rundqwist (l989) , Integrator windup and how to avoid it . Proe. of Amer. Contc_ Con! pp, 1693- 1698 , Campo, P,J, and M, Morari (1990) , Robust control of processes subject to saturation nonlinearities, Camp.t, chem, £ ngng , 14, 343- 358, Desoer, C,A_ and Y,T, Wang (1980), Linear timeinvariant robust servomec ha nism prob lem : A self-contained exposition . C ontrol and Dynamic System, 16 , 81-129, Doyle, J .C" R ,S, Smith and D ,F, Enns (1987) . Control of plants with input ~:\t ura.ti o n nonlinearities. Proc. oJ A m er. C 011tr. Conf pp . 10341039, Esfandiari , F, and Il.K , Khalil (1992), Output feedback stabilization of full y lincariz able systems, Int, J, of Contc, 56 , 1007- 1037_ Kapoor, N" A,R, Teel and p , Daoutidis (1995.), Guidelines for anti-wind up design for linear systems . .':mbmilled lo Auiomatica.
i'
: e
u:l\ :1 './
i
'.'
Time (sec.)
Time (sec.)
Fig.I. State Feedback Design (solid). PI Controller (dash)
I!
::D:' ' .. Q~
;
0.
,
:,
,
U
:
::,i -/V~~
Time (sec.)
Time (sec.)
Fig.2. Output Feedback Design (solid), PI Controller (dash)
Time
Time
Fig.3. PI Controller 0 - , --,-·_-
_---_-~
Uj
X2
'. Time
Fig.4. State Feedback Design
Kapoor, N " p, Daoutidis and A.R, Teel (1995b), Anti-windup with guaranteed stability for linear systems with input. con8iraints. Annual AIChE Mee/mg, paper' no, 178g. Teel, A.R, and L. Praly (1995), Tools for semi, global stabilization by partial state and output feedback, SIAM j, of Conir, and Optimlz. 33,1443-1488. Walgama, K.S. and J, Sternby (1990), Inherent observer property in a class of anti-windup compensators. Int. J. ContT'. 52, 705-724.
1517
Time
_ _ ____,